qLet be a state space realization of a
(not necessarily stable) transfer matrix G(s). Suppose that there exists a symmetric matrix
q
• (P is not necessarily the
Gramian) with P1 nonsingular such that
•AP+PA*+BB*=0
• Now partition the
realization (A,B,C,D) compatibly with P
as
•
•
• Then is also a
realization of G.
• Moreover, (A11,B1) is controllable if A11 is
stable.